Definition 7.1.7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ and $K$ be simplicial sets, and suppose we are given a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$. We say that $\overline{f}$ is a levelwise colimit diagram if, for every vertex $b \in B$, the composite map
\[ K^{\triangleright } \xrightarrow {\overline{f}} \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b} \operatorname{Fun}( \{ b\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]
is a colimit diagram in $\operatorname{\mathcal{C}}$. Similarly, we say that a diagram $\overline{g}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is a levelwise limit diagram if, for every vertex $b \in B$, the composition $\operatorname{ev}_{b} \circ \overline{g}$ is a limit diagram in $\operatorname{\mathcal{C}}$.